Logic Seminar

A pointwise ergodic theorem for the action of a countable group Gamma on a probability space equates ergodicity of the action to its a.e. pointwise combinatorics. One result (joint work with Jon Boretsky) we will discuss is a short, combinatorial proof of the pointwise ergodic theorem for free, probability measure preserving (pmp) actions of amenable groups along Tempelman Følner sequences, which is a slightly less general version of Lindenstrauss's celebrated theorem for tempered Følner sequences. In fact, we prove that such actions have a certain tiling property, which implies the pointwise ergodic theorem. Another result we will discuss is a similar tiling property in the quasi-pmp setting for the natural boundary actions of the free group on n generators (n < infty), which implies the corresponding pointwise ergodic theorem.